Search results for "bifurcation"
showing 10 items of 204 documents
Bifurcation of traveling waves in a Keller–Segel type free boundary model of cell motility
2018
We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) presence of the cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from a family of radially symmetric steady states. The traveling waves describe persistent motion without external cues or stimuli which is a signature of …
Axisymmetric solutions for a chemotaxis model of Multiple Sclerosis
2018
In this paper we study radially symmetric solutions for our recently proposed reaction–diffusion–chemotaxis model of Multiple Sclerosis. Through a weakly nonlinear expansion we classify the bifurcation at the onset and derive the amplitude equations ruling the formation of concentric demyelinating patterns which reproduce the concentric layers observed in Balò sclerosis and in the early phase of Multiple Sclerosis. We present numerical simulations which illustrate and fit the analytical results.
On the stability of bifurcation branches in thermal ignition
1984
A method is given to determine the stability of stationary solutions of the thermal ignition equation for the case ofn-dimensional spherical symmetry, together with the number of unstable modes. For sufficiently high temperature and activation temperature this number is arbitrarily large. Some numerical results on the solutions and their stability are reported.
Lusternik-Schnirelmann Critical Values and Bifurcation Problems
1987
We present a method to calculate bifurcation branches for nonlinear two point boundary value problems of the following type $$ \{ _{u(a) = u(b) = 0,}^{ - u'' = \lambda G'(u)} $$ (1.1) where G : R → R is a smooth mapping. This problem can be formulated equivalently as $$ g' \left(u \right)= \mu u, $$ (1.2) where $$ g \left(u \right)= \overset{b} {\underset{a} {\int}} G \left(u \left(t \right) \right) dt $$ (1.3) and μ = 1/λ. Solutions of this problem can be found by locating the critical points of the functional g : H → R on the spheres \(S_r= \lbrace x \in H \mid \;\parallel x \parallel =r \rbrace, r >0.\) (The Lagrange multiplier theorem.)
Intermittent-Type Chaos in Nonsinusoidal Driven Oscillators
2000
The intermittent-type chaos occurring in rf- and dc- nonsinusoidal driven oscillators is investigated analytically and numerically. The attention is focused on a general class of oscillators in which the total potential VRP(,r) is the Remoissenet-Peyrard potential which has constant amplitude and is 2π-periodic in , and whose shape can be varied as a function of parameter r ( |r| < 1). A simple physical model for calculating analytically the Melnikov function is proposed. The onset of chaos is studied through an analysis of the phase space, a construction of the bifurcation diagram and a computation of the Lyapunov exponent. The parameter regions of chaotic behaviour predicted by the theore…
Domain wall dynamics in an optical Kerr cavity
2004
An anisotropic (dichroic) optical cavity containing a self-focusing Kerr medium is shown to display a bifurcation between static --Ising-- and moving --Bloch-- domain walls, the so-called nonequilibrium Ising-Bloch transition (NIB). Bloch walls can show regular or irregular temporal behaviour, in particular, bursting and spiking. These phenomena are interpreted in terms of the spatio-temporal dynamics of the extended patterns connected by the wall, which display complex dynamical behaviour as well. Domain wall interaction, including the formation of bound states is also addressed.
Experimental approach to transverse wave-number selection in cavity nonlinear optics
2004
Spontaneous transverse pattern formation is experimentally studied in a ${\text{BaTiO}}_{3}$ photorefractive oscillator under degenerate four-wave mixing conditions. A near self-imaging resonator of high Fresnel number and quasi-one-dimensional in the transverse plane is used. A fine control technique of the cavity detuning, $\ensuremath{\Omega}$, is described. It allows a precise study of the relation of $\ensuremath{\Omega}$ with the transverse wave number ${k}_{\ensuremath{\perp}}$ of the roll patterns selected by the system. The law ${k}_{\ensuremath{\perp}}^{2}=\ensuremath{-}\ensuremath{\Omega}∕a$ is verified, which evidences that wave-number selection is mainly dictated by the cavity …
Rich dynamics and anticontrol of extinction in a prey-predator system
2019
This paper reveals some new and rich dynamics of a two-dimensional prey-predator system and to anticontrol the extinction of one of the species. For a particular value of the bifurcation parameter, one of the system variable dynamics is going to extinct, while another remains chaotic. To prevent the extinction, a simple anticontrol algorithm is applied so that the system orbits can escape from the vanishing trap. As the bifurcation parameter increases, the system presents quasiperiodic, stable, chaotic and also hyperchaotic orbits. Some of the chaotic attractors are Kaplan-Yorke type, in the sense that the sum of its Lyapunov exponents is positive. Also, atypically for undriven discrete sys…
Transition to turbulence in toroidal pipes
2011
AbstractIncompressible flow in toroidal pipes of circular cross-section was investigated by three-dimensional, time-dependent numerical simulations using a finite volume method. The computational domain included a whole torus and was discretized by up to ${\ensuremath{\sim} }11. 4\ensuremath{\times} 1{0}^{6} $ nodes. Two curvatures $\delta $ (radius of the cross-section/radius of the torus), namely 0.3 and 0.1, were examined; a streamwise forcing term was imposed, and its magnitude was made to vary so that the bulk Reynolds number ranged between ${\ensuremath{\sim} }3500$ and ${\ensuremath{\sim} }14\hspace{0.167em} 700$. The results were processed by different techniques in order to confirm…
Generalization of the Lorenz-Haken model to atomic systems with different relaxation rates for the two laser levels
1995
Abstract The fundamental Lorenz-Haken laser model is generalized to the case of a two-level amplifying medium with different external relaxation rates for the two levels and with internal relaxation. This represents one further degree of freedom, and important quantitative differences in the laser dynamics. i.e., in the stationary solutions, linear stability analysis, and timedependent solutions, are found. No significant qualitative differences, however, are observed.